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# A disk of radius 0.2 m and moment of inertia $1.5 \mathrm{kg} \cdot \mathrm{m}^{2}$ is mounted on a nearly frictionless axle. A string is wrapped tightly around the disk, and you pull on the string with a constant force of 25 N. After a while the disk has reached an angular speed of 2 rad/s. What is its angular speed 0.1 seconds later? Explain briefly.

Solution

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$\underline{\text{Identify the unknown:}}$

The final angular speed

$\underline{\text{List the Knowns:}}$

Initial angular speed: $\omega_i= 2 \;\mathrm{rad/s}$

Applied force: $F_s= 25 \;\mathrm{N}$

Time interval: $\Delta t = 0.1 \;\mathrm{s}$

Radius of the disk: $R= 0.2 \;\mathrm{m}$

Moment of inertia of the disk: $I=1.5 \;\mathrm{kg \cdot m^2}$

$\underline{\text{Set Up the Problem:}}$

Angular momentum in terms of moment of inertia:

$\vec{L}_{A} = I \vec{\omega}$

$\vec{\tau}_A = \vec{r}_A \times \vec{F}$

The only force that exerts a torque is the force by the string, and the force is applied perpendicular to the position vector, thus the torque is maximum, so:

$\tau = r F= RF_{s}$

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